On the static and dynamic analysis of inflated hyperelastic circular membranes

This paper considers certain aspects of static and dynamic analysis of inflated unstretched and prestretched flat circular hyperelastic membranes. The problem is both geometrically and materially nonlinear. The governing equations of equilibrium, and the equations of small amplitude dynamics are obtained using the variational formulation. The equilibrium configuration of the membrane is obtained by solving a two-point boundary value problem exploiting a scaling symmetry of the equations of equilibrium. Interestingly, in certain cases, beyond a certain inflation of the membrane, the Gaussian curvature flips sign (positive to negative) near the periphery of the membrane leading to neck formation and impending wrinkling condition. The dynamics of perturbations over the static shape have been studied considering both constant pressure and adiabatic conditions. Two remarkable new phenomena, namely a symmetry breaking torsional mode instability via a supercritical pitchfork bifurcation, and a stretch induced softening behaviour of the membrane, are revealed through the analysis.